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Minimization and Mountain-Pass Theorems

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An Introduction to Minimax Theorems and Their Applications to Differential Equations

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 52))

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Abstract

In this introductory chapter, we consider the concept on differentiability of mappings in Banach spaces, Fréchet and Gâteaux derivatives, secondorder derivatives and general minimization theorems. Variational principles of Ekeland [Ek1] and Borwein & Preiss [BP] are proved and relations to the minimization problem are given. Deformation lemmata, Palais—Smale conditions and mountain-pass theorems are considered. The deformation approach and ε—variational approach are applied to prove the mountainpass theorem and its various extensions.

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Author information

Authors and Affiliations

  1. ISEG, Universidade Técnica de Lisboa, Portugal

    Maria do Rosário Grossinho

  2. CMAF, Universidade de Lisboa, Portugal

    Maria do Rosário Grossinho

  3. University of Rousse, Bulgaria

    Stepan Agop Tersian

Authors
  1. Maria do Rosário Grossinho
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  2. Stepan Agop Tersian
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© 2001 Springer Science+Business Media Dordrecht

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do Rosário Grossinho, M., Tersian, S.A. (2001). Minimization and Mountain-Pass Theorems. In: An Introduction to Minimax Theorems and Their Applications to Differential Equations. Nonconvex Optimization and Its Applications, vol 52. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3308-2_1

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  • DOI: https://doi.org/10.1007/978-1-4757-3308-2_1

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4849-6

  • Online ISBN: 978-1-4757-3308-2

  • eBook Packages: Springer Book Archive

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